Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

Abstract

In this article, we study the existence of renormalized solution for the nonlinear $p(x)$-parabolic problem of the form:\\ $\begin{cases} \frac{\partial b(x,u)}{\partial t} - div (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - divF $ \; in $Q= \Omega\times(0,T)\\$ $b(x,u)\mid_{t=0} = b(x,u_{0})$ \; in $\Omega\\$ $ u = 0 $ \quad on $\partial\Omega\times(0,T)\\$ $\end{cases}$ with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $\\ The main contribution of our work is to prove the existence of renormalized solution of the variable exponent Soblev spaces, and we suppose that\;$ H(x,t,u,\nabla u)$\; is the non linear term satisfying some growth condition but no sig

Keywords

• Variable exponent Sobolev
• Young’s Inequality
• Renomalized Solution
• Parabolic problems
• Tree unbounded nonlinearities